Validation of an axially distributed model for quantification of myocardial blood flow using ¹³N-ammonia PET

Abstract

Background

Estimation of myocardial blood flow (MBF) with cardiac PET is often performed with conventional compartmental models. In this study, we developed and evaluated a physiologically and anatomically realistic axially distributed model. Unlike compartmental models, this axially distributed approach models both the temporal and the spatial gradients in uptake and retention along the capillary.

Methods

We validated PET-derived flow estimates with microsphere studies in 19 (9 rest, 10 stress) studies in five dogs. The radiotracer, ¹³N-ammonia, was injected intravenously while microspheres were administered into the left atrium. A regional reduction in hyperemic flow was forced by an external occluder in five of the stress studies. The flow estimates from the axially distributed model were compared with estimates from conventional compartmental models.

Results

The mean difference between microspheres and the axially distributed blood flow estimates in each of the 17 segments was 0.03 mL/g/minute (95% CI [−0.05, 0.11]). The blood flow estimates were highly correlated with each regional microsphere value for the axially distributed model (y = 0.98x + 0.06 mL/g/minute; r = 0.74; P < .001), for the two-compartment (y = 0.64x + 0.34; r = 0.74; P < .001), and for three-compartment model (y = 0.69x + 0.54; r = 0.74; P < .001). The variance of the error of the estimates is higher with the axially distributed model than the compartmental models (1.7 [1.3, 2.1] times higher).

Conclusion

The proposed axially distributed model provided accurate regional estimates of MBF. The axially distributed model estimated blood flow with more accuracy, but less precision, than the evaluated compartmental models.

Appendix

Axially Distributed Model

The input function TAC was taken from a region of interest in the left atrium. In order to account for the delay from this site and the coronary arterial inflow to the individual ROI, the tracer traversed a dispersive convective tube of length L _z given by

$$ \frac{{V_{\text{cav}} \partial C_{\text{N,cav}} }}{\partial t} = - F_{\text{p}} L_{z} \frac{{\partial C_{\text{N,cav}} }}{\partial z} + V_{\text{cav}} D_{\text{cav}} \frac{{\partial^{2} C_{\text{N,cav}} }}{{\partial z^{2} }}. $$

This relationship allows for a finite time for passage from the input function ROI (the first term on the right) and realistic dispersion of the signal (the second term) and is optimized for each blood flow estimate by estimating F _p and V _cav. The initial condition for the concentration in the unit is zero. The boundary condition at the outflow is simple reflection so there is no diffusion into the outflow, only convection, $\partial C_{\text{N,cav}} /\partial z = 0$, and $C_{\text{out}} = C_{\text{N,cav}} (z = L)$. At the inflow, the boundary condition is such that the diffusion upstream and the convection downstream are matched: $F_{\text{p}} L \cdot {{\left( {C_{\text{in}} - C_{\text{N,cav}} } \right)} \mathord{\left/ {\vphantom {{\left( {C_{\text{in}} - C_{\text{N,cav}} } \right)} {{{V_{\text{lv}} + D_{\text{cav}} \cdot \partial C_{\text{N,cav}} } \mathord{\left/ {\vphantom {{V_{\text{lv}} + D_{\text{cav}} \cdot \partial C_{\text{N,cav}} } {\partial z}}} \right. \kern-0pt} {\partial z}}}}} \right. \kern-0pt} {{{V_{\text{lv}} + D_{\text{cav}} \cdot \partial C_{\text{N,cav}} } \mathord{\left/ {\vphantom {{V_{\text{lv}} + D_{\text{cav}} \cdot \partial C_{\text{N,cav}} } {\partial z}}} \right. \kern-0pt} {\partial z}}}} = 0$, where C _in, is the input function.

The major components for the two-region unit are represented graphically in Figure 2. The regional concentrations of ¹³N-NH₃ and ¹³N-glutamine in the plasma are each defined in terms of mol/mL and are functions of capillary axial position, x, and of time, t. For example, the partial differential equation for the concentration of NH₃ in the plasma, $C_{\text{N,p}},$ is

$$ \frac{{V_{\text{p}} \partial C_{\text{N,p}} }}{\partial t} = - F_{\text{p}} L\frac{{\partial C_{\text{N,p}} }}{\partial x} + {\text{PS}}_{\text{N}} \left( {C_{\text{N,m}} - C_{\text{N,p}} } \right) + V_{\text{p}} D_{\text{p}} \frac{{\partial^{2} C_{\text{N,p}} }}{{\partial x^{2} }} $$

(1)

and for NH₃ in the myocyte, $C_{\text{N,m}}$, is

$$ \frac{{V_{\text{m}} \partial C_{\text{N,m}} }}{\partial t} = {\text{PS}}_{\text{N}} \left( {C_{\text{N,p}} - C_{\text{N,m}} } \right) - GC_{\text{N,m}} + V_{\text{m}} D_{\text{m}} \frac{{\partial^{2} C_{\text{N,m}} }}{{\partial x^{2} }} $$

(2)

The boundary conditions in the capillary are analogous to those in the delay line above, and in the tissue are reflecting boundaries at x = 0 and x = L. The concentration of glutamine in the plasma and myocyte follows similar governing equations as (1) and (2) and for trapped glutamate in the myocyte as (2). The plasma region has the known initial boundary condition of the input function, C _N,cav(z = L) and the output boundary condition of C _out. The total ¹³N-ammonia in a region of interest at any given time, t, is measured in mol/g as

$$ Q_{\text{tot}} = V_{\text{p}} \int\limits_{0}^{L} {\left( {C_{\text{N,p}} + C_{\text{MI,p}} } \right)\,{\text{d}}x} + V_{\text{m}} \int\limits_{0}^{L} {\left( {C_{\text{N,m}} + C_{\text{MI,m}} + C_{\text{MA,m}} } \right)\,{\text{d}}x + {\text{spill}} \times C_{\text{N,lvout}} } , $$

where spill (mL/g) is the fraction of the delayed input function that has spilled over into the myocardial ROI. This model is available for download and independent testing at http://www.physiome.org/jsim/.